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# Hegel on Mathematical Infinity

The concept of infinity plays a pivotal role in mathematics, yet its precise definition remains elusive. This conceptual ambiguity has given rise to several puzzles in contemporary philosophy of mathematics. In response, this dissertation embarks on a rational reconstruction of Hegels concept of infinity and applies it to resolve two groups of mathematical puzzles, including challenges in applied mathematics, especially the application of differential calculus, and the conceptual ground of set theory, especially Cantors paradox.

The exploration begins with a historical survey of the concept of infinity in philosophy. It becomes evident that a prevailing interpretation characterizes infinity as the unlimited. In addition, this unlimitedness has taken various forms, including endlessness (Aristotle), all-inclusiveness (Spinoza), and self-sufficiency (Kant).

The heart of the dissertation lies in reconstructing Hegels concept of genuine infinity. Hegel argues that the unlimited as the negation of the limit entails either the completely indeterminate or another limited entity, neither of which is genuinely infinite. Instead, Hegel points out that genuine infinity is the self-relation of a limited entity. By self-relation, Hegel means that the limited entity alters into another limited entity that is isomorphic to the original one.

Subsequently, Hegel’s concept of genuine infinity can be translated into a mathematical framework as the intrinsic alteration of quantum (roughly speaking, quantum is Hegel’s term for the variable), which is captured by the corresponding relation among quanta. It is argued that this relation serves as the necessary condition for three mathematical entities traditionally considered infinite: arbitrarily large (small) numbers, infinite sets, and endless sequences. Thus, for Hegel, this intrinsic relation among quanta constitutes the essence of mathematical infinity.

Hegels concept of mathematical infinity can help us resolve difficulties within contemporary mathematics. First, it addresses the question of why infinite mathematical structures can be applied to describe and predict seemingly finite physical phenomena. The application of mathematics is usually explained by the similarity between mathematical structures and empirical systems, but the lack of apparent empirical counterpart leads one to doubt the application of infinite mathematical structures. Hegels concept of mathematical infinity directs us to focus on the structural similarity between infinite mathematical structures and empirical systems, specifically between the intrinsic alteration of quantum and the change of physical properties with time. With this structural similarity, the application of mathematics can be explained. Second, the dissertation investigates the conceptual ground of set theory, especially the relationship between a set and its members. Hegels analysis of genuine infinity provides a twofold clarification: (1) members of set must be a unit first, which entails that the set of all sets (the Universe) is not a set; (2) members of a set are simultaneously distinct (due to their independent logical content) yet indistinguishable (due to their common structure as a unit). Clarification 1 resolves Cantors paradox as it excludes the Universe; clarification 2 explains arithmetic operations.

## History

## Degree Type

- Doctor of Philosophy

## Department

- Philosophy

## Campus location

- West Lafayette